# Thread: Integration with Euler's constant and cosine

1. ## Integration with Euler's constant and cosine

Integrate

e^(2x)cosx dx

I tried substitution, where u=2x, but then I am still left with an x value from cosx. I know how to integrate them separately, but now when they are multipled together... how should I go about integrating?

2. How's your "by parts"? Do it twice.

3. Originally Posted by bhuang
Integrate

e^(2x)cosx dx

I tried substitution, where u=2x, but then I am still left with an x value from cosx. I know how to integrate them separately, but now when they are multipled together... how should I go about integrating?
You need to use integration by parts.

$\displaystyle \int{u\,dv} = u\,v - \int{v\,du}$.

So for $\displaystyle \int{e^{2x}\cos{x}\,dx}$

Let $\displaystyle u = e^{2x}$ so that $\displaystyle du = 2e^{2x}$.

Let $\displaystyle dv = \cos{x}$ so that $\displaystyle v = \sin{x}$.

Therefore $\displaystyle \int{e^{2x}\cos{x}\,dx} = e^{2x}\sin{x} - \int{2e^{2x}\sin{x}\,dx}$

$\displaystyle = e^{2x}\sin{x} - 2\int{e^{2x}\sin{x}\,dx}$.

Now you need integration by parts again.

Let $\displaystyle u = e^{2x}$ so that $\displaystyle du = 2e^{2x}$.

Let $\displaystyle dv = \sin{x}$ so that $\displaystyle v = -\cos{x}$.

So $\displaystyle e^{2x}\sin{x} - 2\int{e^{2x}\sin{x}\,dx} = e^{2x}\sin{x} - 2\left[-e^{2x}\cos{x} - \int{-2e^{2x}\cos{x}\,dx}\right]$

$\displaystyle = e^{2x}\sin{x} + 2e^{2x}\cos{x} - 4\int{e^{2x}\cos{x}\,dx}$.

So we have

$\displaystyle \int{e^{2x}\cos{x}\,dx} = e^{2x}\sin{x} + 2e^{2x}\cos{x} - 4\int{e^{2x}\cos{x}\,dx}$

$\displaystyle 5\int{e^{2x}\cos{x}\,dx} = e^{2x}\sin{x} + 2e^{2x}\cos{x}$

$\displaystyle \int{e^{2x}\cos{x}\,dx} = \frac{1}{5}e^{2x}\sin{x} + \frac{2}{5}e^{2x}\cos{x} + C$.